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Integer and rational exponents — ACT practice questions

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Practice 41 Integer and rational exponents questions in the app

What this topic is

Exponent rules form the core of Integer and rational exponents.

Multiplying powers with the same base adds the exponents, raising a power to a power multiplies them, a negative exponent means a reciprocal, and a fractional exponent means a root, so eight to the one-third power equals two. A frequent mistake is adding exponents when the bases differ. ACT Mathematics asks for equivalent expressions, compares the sizes of several powers, or works with scientific notation, where numbers are written as a value times a power of ten.

Answer choices tend to be close variations, such as the same base with different exponents, so careful rule-by-rule simplification pays off.

Sample questions

Pick an answer to see whether it is right — nothing is saved, and nothing needs an account.

Question 1Easier

Which of the following is equivalent to $\frac{18 u^{12} v^{5}}{3 u^{4} v^{2}}$ for all nonzero $u$ and $v$ ?

6 u^{3} v^{3}

✗ Not this one

15 u^{8} v^{3}

✗ Not this one

6 u^{16} v^{7}

✗ Not this one

6 u^{8} v^{3}

✓ Correct

Show answer

Answer: D — $6 u^{8} v^{3}$

The coefficient is $18 \div 3 = 6$ . Subtracting exponents gives $u^{12 - 4} v^{5 - 2} = u^{8} v^{3}$ , so the ratio is $6 u^{8} v^{3}$ .

Question 2Easier

Which of the following is the greatest?

9^{- 1 / 2}

✗ Not this one

8^{- 1 / 3}

✓ Correct

1 6^{- 1}

✗ Not this one

2 5^{- 1 / 2}

✗ Not this one

Show answer

Answer: B — $8^{- 1 / 3}$

The four values are $\frac{1}{3}$ , $\frac{1}{2}$ , $\frac{1}{16}$ , and $\frac{1}{5}$ , respectively. The greatest is $8^{- 1 / 3} = \frac{1}{2}$ .

Question 3Easier

A reservoir initially contains $4.8 \times 1 0^{6}$ liters of water. Water drains at a constant rate of $7.5 \times 1 0^{4}$ liters per hour for 8 hours. How many liters remain, expressed in scientific notation?

4.725 \times 1 0^{6}

✗ Not this one

5.4 \times 1 0^{6}

✗ Not this one

4.2 \times 1 0^{6}

✓ Correct

4.2 \times 1 0^{5}

✗ Not this one

Show answer

Answer: C — $4.2 \times 1 0^{6}$

The amount drained is $(7.5 \times 1 0^{4}) (8) = 6 \times 1 0^{5}$ liters. Rewriting this as $0.6 \times 1 0^{6}$ gives $(4.8 - 0.6) \times 1 0^{6} = 4.2 \times 1 0^{6}$ liters remaining.

Every question is picked for where you are right now, wrong answers come back until they stick, and it all works offline.

Practice 41 Integer and rational exponents questions in the app

Integer and rational exponents — ACT practice questions

What this topic is

Sample questions

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Mathematics